Step by step
The question
Question: solve x² − 4x = 12 by completing the square.
Is x alone now?
Not yet: − 4 is still with x.
How do we remove it?
1 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
x² is a square: x by x.
−4x is cut into two equal parts: one beside the square, one below it.
With a minus the picture is harder, but the numbers work the same.
Half of −4 is −2, so each part is −2 times x.
One corner is missing: −2 times −2.
Minus times minus is plus, so it is 4.
With it, the grid is full.
So we add 4 to fill the corner.
We do it on both sides, so they stay equal.
Careful!
Many people add 4 only on the left.
Then the two sides are not equal any more.
2 Write it as a square
The full square is x − 2 on each side.
So x² − 4x + 4 is (x − 2)².
12 plus 4 is 16.
Now we have (x − 2)² = 16.
3 Square root of both sides
We want x alone.
But here we have (x − 2)².
(x − 2)² means (x − 2) times (x − 2).
So we need to remove the ².
To remove a square, we take the square root.
The square root of 16 is 4, because 4² = 16.
But (−4)² = 16 too.
Minus times minus is plus.
So there are two answers: plus and minus.
x − 2 = ±4.
That means x − 2 = 4 or x − 2 = −4.
So after step 3: we took the square root, and now we have x − 2 = 4 or x − 2 = −4.
4 Get x alone
To remove the − 2, we add 2 on both sides.
We do it in both equations.
4 plus 2 is 6.
Minus 4 plus 2 is −2.
So x = 6 or x = −2.
5 Check
Let's check: put x = 6 into the left side.
We get 12, the same as the right side.
It works!
Now put x = −2 into the left side.
We get 12, the same as the right side.
So both answers are right!
Let's look back.
First: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.